The Segal Algebra {\bf S}_0({\Bbb R}^d) and Norm Summability of Fourier Series and Fourier Transforms

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier series. Equivalent conditions are derived for the uniform and L ₁-norm convergence of the θ-means σ _n ^θ f to the function f. If f is in a homogeneous Banach space, then the preceeding convergence holds in the norm of the space. In case θ is an element of Feichtinger’s Segal algebra , then these convergence results hold. Some new sufficient conditions are given for θ to be in . A long list of concrete special cases of the θ-summation is listed. The same results are also provided in the context of Fourier transforms, indicating how proofs have to be changed in this case. This research was supported by Lise Meitner fellowship No M733-N04 and the Hungarian Scientific Research Funds (OTKA) No T043769, T047128, T047132.     

Universal method for synthesis of artificial gel antibodies by the imprinting approach combined with a unique electrophoresis technique for detection of minute structural differences of proteins, viruses, and cells (bacteria): II. Gel antibodies against virus (Semliki Forest Virus)

Artificial and highly selective antibodies (in the form of gel granules) against proteins can easily be synthesized by a simple, cost-effective imprinting technique [Liao, J.-L. et al., Chromatographia 1996, 42, 259-262]. Using the same method for synthesis of gel antibodies against viruses in combination with analysis by free zone electrophoresis in a rotating narrow bore tube we have shown that artificial gel antibodies against Semliki Forest Virus (wild type) can sense the difference between this virus and a mutant, although they differ in their chemical composition only by three amino acids in one of the three proteins on the surface of the virus particle. The reason for this extremely high resolution is explained by the fact that we use three types of selectivity: (i) shape selectivity (created by the close fit between the antigen and its imprint in the gel), (ii) bond selectivity in the contact area between the antigen and its imprint in the gel antibody, and (iii) charge selectivity, originating from slightly different structures or/and conformations of the antigens.     

Simulated vibrational spectra of aflatoxins and their demethylated products and the estimation of the energies of the demethylation reactions

The structure of four natural mycotoxins, the aflatoxin B1, B2, G1 and G2 and their demethylated products were optimized with quantum chemical method. The energies and the thermodynamic functions of the molecules were calculated and applied to calculation of the reaction energies of the demethylations. Further results of the calculations are the vibrational force constants, the infrared spectra of the molecules and the assignments of the spectral bands.     

On the problem of uniqueness of the trigonometric moment constants

Summary Given a real-valued function <InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mu(x,y)$ of bounded variation in the sense of Hardy and Krause on the square $[0, 2\pi]\times [0, 2\pi]$, the sequence <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \mu_{m,n}:=\int^{2\pi}_0 \int^{2\pi}_0 e^{i(mx+ny)} \, d_x \, d_y \mu(x,y), \quad (m,n)\in \bZ^2, $$ may be called the sequence of trigonometric moment constants with respect to $\mu(x,y)$. We discuss the uniqueness of the expression of the sequence $\{\mu_{m,n}\}$ in terms of the function $\mu(x,y)$.     

Simulations of petawatt-class few-cycle optical-parametric chirped-pulse amplification, including nonlinear refractive index effects

We present three-dimensional simulations of optical-parametric chirped-pulse amplification stages for a few-cycle petawatt-class laser. The simulations take into account the effects of depletion, diffraction, walk-off, quantum noise, and the nonlinear refractive index (n(2)). In the absence of n(2) effects, we show these stages can generate 3.67J pulses supporting 4fs transform-limited pulse durations. Adding the nonlinear refractive index to the simulation, the energy output is reduced by ~11% and the bandwidth narrows by ~129nm, increasing the Fourier limit by ~17.5%.     

Synthesis and application of a fluorous Lawesson's reagent: convenient chromatography-free product purification

[reaction: see text] A fluorous analogue of Lawesson's reagent for thionation of carbonyl compounds has been developed and its use demonstrated on a series of amides, esters, and ketones. The separation of the Lawesson's reagent-derived byproducts can be achieved by a simple fluorous solid-phase extraction.     

Absolute convergence of Walsh-Fourier series and related results

We consider the Walsh orthonormal system on the interval [0, 1) in the Paley enumeration and the Walsh-Fourier coefficients $ \hat f $ (n), n ∈ ?, of functions f ∈ L ^p for some 1 < p ≤ 2. Our aim is to find best possible sufficient conditions for the finiteness of the series Σ _n=1 ^∞ a _n | $ \hat f $ (n)| ^r , where {a _n } is a given sequence of nonnegative real numbers satisfying a mild assumption and 0 < r < 2. These sufficient conditions are in terms of (either global or local) dyadic moduli of continuity of f. The sufficient conditions presented in the monograph [2] are special cases of our ones.